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t!'hrons # I ' q g(x''6) *r' ('€r),.-' it l=5 ., x,! posihve Iltcgers O Ol'lPt"a 's'<- =1 No|e H^cd o!4c db $'u dtee R l' t* P&rr-J i '' j, 'b tt* l''lts ' So n?1 Y3 rL, Y=n3 B 11-:b^ c.le .\ 't{4 2S ot-f"qr Q"g"d^ n.[t) {'{^, 1'wt vn-l difl/\ ^ '!' o cvttsA' ctvL b) ?y rl) : 6)'-'ti,) 0 (1, r) -, fiP) '(t,3) , ( t,'r), (l,s), (t,a) (z,r) #r[s.r))t4,t)., ( s, \) (.a,t) ro\ls ocLLti- ( cw"^'e.. t V" f.- fo\ts, Y1*"- @' owe "1, fu U rolls nl{^ ,-ull , $ posihr're- OVl"g.rW r\-€ i nEger c) E fY *zY) a- Smce X= Y r E [x*zY] ' E (X] * ?EH) t"( We" 'rt 7 3x (4)'.(=i') {tt ; 3 ., [%1)"-'[;') *=--/ A*Yxa voJd't 3 fl,1 =7L J \ftl f ":3;'*r.^'-re na 'Tnc ar ftu.,t^ tl 'r tro*, r'V' wiF Y" 76 I > J I' L, (") .rt'+J ='h- --' - Y =r [b) n,,f dx/4 2gdt = L &u d'1 =t'/ rl Is..r'c 5JJ ' I'f' oO f\J 0tx [.) {*r"r' J-i-t-q)dY - Ji {,", ttt = ti+"''id* - 5: (d) Noc +(,, t ) -- o erwe'o < 1--2x o<xcl Z dt ' Z- Z* ' frf'-l' 1 o 6\'t^t<v'stv- zqf,d* = z5 s) {Y','''1? :ffi:,. tr\, B,,t t*tl) = \ a^d tyf gt' + ; tct,'^) ;^8r,'t)ttt{) => NirrurygP^ I
Transcript
Page 1: math.gmu.edumath.gmu.edu/~rgoldin/351/Sol8.pdf · Created Date: 12/9/2012 8:57:07 PM

t!'hrons # I

' q g(x''6) *r' ('€r),.-' it l=5 ., x,! posihve Iltcgers

O Ol'lPt"a 's'<-

=1

No|e H^cd o!4c db $'u dtee R l' t* P&rr-J

i '' j, 'b tt* l''lts '

So n?1 Y3 rL, Y=n3 B

11-:b^ c.le .\ 't{4 2S ot-f"qr

Q"g"d^ n.[t) {'{^, 1'wt vn-l

difl/\ ^ '!' o cvttsA' ctvL

b) ?y rl) : 6)'-'ti,)0

(1, r) -, fiP) '(t,3) , ( t,'r), (l,s),(t,a)

(z,r) #r[s.r))t4,t).,( s, \) (.a,t)

ro\ls ocLLti- ( cw"^'e.. t V" f.-fo\ts, Y1*"- @' owe

"1, fu U rolls

nl{^ ,-ull ,

$ posihr're-

OVl"g.rW r\-€

i nEger

c) E fY *zY) a-

Smce X= Y r

E [x*zY] '

E (X] * ?EH)t"( We" 'rt

7 3x (4)'.(=i'){tt

; 3 ., [%1)"-'[;')*=--/A*Yxa voJd't

3 fl,1 =7LJ \ftl f

":3;'*r.^'-re na'Tnc arftu.,t^

tl 'r

tro*, r'V' wiF Y" 76 I > J

I'

L, (") .rt'+J='h- --' - Y =r [b)

n,,f dx/4

2gdt =

L &u d'1

=t'/rl Is..r'c5JJ

' I'f'oO

f\J0tx

[.) {*r"r' J-i-t-q)dY - Ji

{,", ttt = ti+"''id* -

5:

(d) Noc +(,, t ) -- o erwe'o

< 1--2x o<xcl

Z dt ' Z- Z* ' frf'-l' 1 o 6\'t^t<v'stv-

zqf,d* = z5 s) {Y','''1? :ffi:,.tr\, B,,t t*tl) = \ a^d tyf gt' +; tct,'^) ;^8r,'t)ttt{) => NirrurygP^

I

Page 2: math.gmu.edumath.gmu.edu/~rgoldin/351/Sol8.pdf · Created Date: 12/9/2012 8:57:07 PM

. ztz= xx,7l

$to

3,

ty, ,\'I-7(tx,s)tt^= J'*ro* =

-) f ,,t)' ?tt o'J'2tY'= to p.,y',

a) fxf ,)= 5: fu,,,1)dy= J i "ro,:) $rft)=1 >-r o4xel

I o o'!4/'

Wzt

t* "e (o' r)

o lxZrorlc

til,'= )t

Ytr

q

u)

tr G-x)"tvr r('8 )o\' *(4f d*=

tJU 1b

' t [' b'*- zr,v' + *s) ax

'+'g{'t-2"€.4n'= lrt -2 tt*

ft*a $^o pd( ,i''{^{ don^rrc'Jnve )tS

L3

*'l.I = J,' f" rJdsdx ' I x*t;* = J.b"

- q) dt,"\&Y l:a.3 yz- x'llt=\n?r"Ej

c) Lat z =f-,r/:,r *{uaafd:?1-z+"\-- l' t-", de otx t"'4

--

A o.rc w-ta,n 4,,-iz'j=e-\ jhen e z \ o^ot tcss *n^; o Il) !

t""us; \ ff^, -:;;; o.*^*r c

., o t "i'"::

Tr=ry \ '-t"' .t/:: dx

c=tt ,I

f tz)* rz*- t- ! m(zt\

Page 3: math.gmu.edumath.gmu.edu/~rgoldin/351/Sol8.pdf · Created Date: 12/9/2012 8:57:07 PM

o(ccl,

x"c \=c4

tJ,t 5'b

| !=Z*( *"Jo^

)r= 2 -!+J L.t'J

tb

Y4*+4"

w

c -1a*h?

^J otf dx

JxL

vr=J-Jz

3@) orn}.

rli G'i

LrneS fr- s4c ( \

1- o-xr>

c

-J J" t< cc- rt a^

'-; e: -z'f * ?"/,

'i(.; Z.'*i)i l+-+,',)"n = i(+-Y"r

' +U') cu '

tnwn,-{ j'. j xJ d-tv

OH7

T- *ru;a t"lk tt *"Ao"A'tv^s '

-tilif is itt &l-trro-rrr'tY[.l,"* 4a << ) '

*Lhy il; fus.lu.,*' ,*r rx{ XNLY,

, lc,tz- /

c,u1c

l"lWst"* lltz

,/-e*^ X =C '

tt-* d {dc 4ro *^"tIt" \",,\F

rr

{"1

Y?xa.3.

{- occ< I

(1 F - Z,H)3

ee&i4^, Cd l' dl

c=rl

tn' c) frie<gcv g'ZJ2

u.I1-'. ar7 , &r g=c-"a , xte-"1

he$ft Xs,m^u Fiu'"tra. ''

@ y4?V"-

Page 4: math.gmu.edumath.gmu.edu/~rgoldin/351/Sol8.pdf · Created Date: 12/9/2012 8:57:07 PM

,.r . "flf;*,f;i r;ur oat4\.,

.' ?7w X<t 8i'r3' 71

cnob Yha*

t t{ +

F,* X a"^or

t"' atcr{)

,'

F'o.r t1D A rt'q@crl| is o-IVw" pd{ q rnnrh [X' I ) =

x<+. 8\>t3 + F1y<t/.X>1iWe t],nrr.e g-nFcowtao

O"l.L l*,r+1"ot\ adu*e).

Plx.rlplY>t3 r ?7Yrl-J,v'tfut*ncb"-rt )

P1x>t7

+tuhY c^rLe

.> iZ*Zt -1.* = Lt -t

Y tr x ,-t3 ?7..l <ts

{'",ce W X a.qclY

+

N1.L

$-ftetnc^,f\w-l'"J t". o at t l,

Pl /'n,nrx,Y).t3 s l-; [-

* &)(l -t )wn i}xr'rt $ clas rt; b*t<J)

Y7x>t LY>L] u

Y1 x > +-jP1 "l > t 3 vtrtflt^dr-"&

I r-L) (r-t) s'ffi W.'-"'"-qMs [n'b^J</

!,O <L<L

Ol.1l,o.'rvrS't kL \

[-

t - (t-zo*&)

- 2t -.tr* ,rlf 1r, frn V7, f' {rr.'1->-$2-z\

= lr to) 1'o-- *b t \)

Page 5: math.gmu.edumath.gmu.edu/~rgoldin/351/Sol8.pdf · Created Date: 12/9/2012 8:57:07 PM

5(a),

Let

P3x >t 3w= x^/,

x-Y >o]Ws ^

r, \(s,'t)Sii.tl,t tl6

cuvr d

=11SrvlCe

v4,Jo lunrn^! f c+'<Atvvt vonic&/La

_y -N(olrrl)'rl

^o, I 'nury 1't'* .vi{:,'

Ec@k

+t"'i"t'*tr <t"U".4,Y,gi nr3,il.1 'ffffi-iltor*v:"tt ri 1' y". t Y ).

boY,w Lw.ral fi - lt /( 5 + L, 4'H)

W-N Ia,l:)

flw>o\ = Y|H >

td P?xr1 7s3-- Y1

t-e+ vrl = XIY

ur>31 = T1+: >

W ^/ N (5-1 , 1+{)ru tq,tj)

a ?1 Zr*'t- > -b \fr3J

x ,q{

5 77Ztta<

Y\ 7,v <

3- "l 1,

-)i,tti

-r 1Jrj)-Lt\f;3 J

,Ll7\ * "boq

Page 6: math.gmu.edumath.gmu.edu/~rgoldin/351/Sol8.pdf · Created Date: 12/9/2012 8:57:07 PM

r4- pUh- %

H^aiv'1 ""^a f eE' hk "v' vton-ryzm

VNI/,!/) wt-o,^ X".g>lO (

s^e' x,' o

7 O 6rw,

fr,ry1 h*t ru cdf, ' F,e ccd)

X*Y 4 f5 'G, uin

P14X rz )-

n t-Y

\ "

Ag'2* s2t ot x d1 = f,* ur,rtil^r,\h)o

,1,, oft ()) (.-t t+'/) -r')n,

.Le'-t, * Z"rr)1:eZL - e^Zt -r l - Tlrr pc( ':

t* cunvc$ve

@'^g'i'|"t;#";-;1;ffil_ r@r,,! /3) cre l=t {*";,, 'llLF L>-rs

^.-r \e g,t)cz-5)

J;ro 4 I

\-zv t = a^--22 I-Y - 7ze

I

wt',-0.,-xrY'tY-=i'" ,[*

ftr,Y) ' fxtx) l'7t1) J o

' Z&n Aet \)v>zot tr:rce X, Y wytfu;c^M

' IJ('

#l4 Z'l*Y

rAt

f.H= )* f*['-. pz -J() Ee^Jo

.: f k,I 4;J,

=) \t"'' ' lze-2'o

Zzogi\**,',:tx

Page 7: math.gmu.edumath.gmu.edu/~rgoldin/351/Sol8.pdf · Created Date: 12/9/2012 8:57:07 PM

(,-(**yrz, du dq d *Zzx*1j) ('* nN

J).)o "6 Fiy

/LeeP '> nnitrd -llne

Pd$s c'\{ 0

l4cLf fo{\n&^ p70,$70

\El/ffi

7, Y1

f^ ror{, t t) lz[")= le'''1e J X, z

ttt"r"'^ YtS t LZ o ,

r4 r'fu

| { €-2(xn)), -J "

t dYax

1,tu^ $+t-z)l dyJ^-

/('Lex+Y

= l^)e-znd* a -l-J, z" 4

l'{"r40\ bg 3 F-

' j: r -e

k*l.ttt?

= _/4X> y +Z

'l z x+z


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